U.S. patent application number 14/055780 was filed with the patent office on 2014-11-13 for apparatus and method for deterministic control of surface figure during full aperture pad polishing.
The applicant listed for this patent is Michael Douglas Feit, William Augustus Steele, Tayyab Ishaq Suratwala. Invention is credited to Michael Douglas Feit, William Augustus Steele, Tayyab Ishaq Suratwala.
Application Number | 20140335767 14/055780 |
Document ID | / |
Family ID | 42184143 |
Filed Date | 2014-11-13 |
United States Patent
Application |
20140335767 |
Kind Code |
A1 |
Suratwala; Tayyab Ishaq ; et
al. |
November 13, 2014 |
APPARATUS AND METHOD FOR DETERMINISTIC CONTROL OF SURFACE FIGURE
DURING FULL APERTURE PAD POLISHING
Abstract
A polishing system configured to polish a lap includes a lap
configured to contact a workpiece for polishing the workpiece; and
a septum configured to contact the lap. The septum has an aperture
formed therein. The radius of the aperture and radius the workpiece
are substantially the same. The aperture and the workpiece have
centers disposed at substantially the same radial distance from a
center of the lap. The aperture is disposed along a first radial
direction from the center of the lap, and the workpiece is disposed
along a second radial direction from the center of the lap. The
first and second radial directions may be opposite directions.
Inventors: |
Suratwala; Tayyab Ishaq;
(Pleasanton, CA) ; Feit; Michael Douglas;
(Livermore, CA) ; Steele; William Augustus;
(Tracy, CA) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
Suratwala; Tayyab Ishaq
Feit; Michael Douglas
Steele; William Augustus |
Pleasanton
Livermore
Tracy |
CA
CA
CA |
US
US
US |
|
|
Family ID: |
42184143 |
Appl. No.: |
14/055780 |
Filed: |
October 16, 2013 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
|
|
12695986 |
Jan 28, 2010 |
8588956 |
|
|
14055780 |
|
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Current U.S.
Class: |
451/59 ;
451/64 |
Current CPC
Class: |
B24B 53/017 20130101;
B24B 37/10 20130101; B24B 53/12 20130101; B24B 37/105 20130101;
B24B 37/04 20130101; B24B 37/32 20130101; B24B 37/042 20130101;
B24B 13/00 20130101 |
Class at
Publication: |
451/59 ;
451/64 |
International
Class: |
B24B 37/04 20060101
B24B037/04; B24B 13/00 20060101 B24B013/00; B24B 37/10 20060101
B24B037/10 |
Goverment Interests
STATEMENT AS TO RIGHTS TO INVENTIONS MADE UNDER FEDERALLY SPONSORED
RESEARCH OR DEVELOPMENT
[0002] The United States Government has rights in this invention
pursuant to Contract No. DE-AC52-07NA27344 between the United
States Department of Energy and Lawrence Livermore National
Security, LLC.
Claims
1. The polishing system of claim Error! Reference source not
found., further comprising the workpiece.
2. A method for pressing a lap with a septum to compress the lap
during polishing of a workpiece to inhibit the workpiece from
compressing the lap during the polishing, the method comprises:
pressing on a workpiece with a first forcing device to place a
first amount of pressure between a lap and a workpiece; and
pressing on a septum with a second forcing device to place a second
amount of pressure between the septum and the lap, wherein the
septum has an aperture formed therein and the workpiece is
configured to contact the lap through the aperture, and wherein the
second amount of pressure is three or more times greater than the
first amount of pressure.
3. The method of claim 2, further comprising rotating the lap with
respect to the septum and the workpiece.
4. A polishing system configured to polish a lap, the polishing
system comprising: a lap configured to contact a workpiece for
polishing the workpiece; and a septum configured to contact the
lap, wherein: the septum has an aperture formed therein; the
aperture has substantially the same radius as the workpiece; the
aperture has a center disposed at a radial distance from a center
of the lap, and disposed along a first radial direction of the lap;
and the workpiece has a center disposed at the radial distance from
the center of the lap, and disposed along a second radial direction
from the center of the lap.
5. The polishing system of claim 4, wherein the septum is
configured to polish the lap to a substantially planar surface as
the lap polishes the workpiece.
6. The polishing system of claim 4, wherein the first radial
direction and the second radial direction are oppositely
directed.
7. The polishing system of claim 4, further comprising the
workpiece.
8. The polishing system of claim 4, wherein the septum has a
substantially triangular shape.
Description
CROSS REFERENCE TO RELATED APPLICATIONS
[0001] This application claims priority to U.S. Provisional Patent
Application No. 61/148,236, filed Jan. 29, 2009, titled
"DETERMINISTIC CONTROL OF SURFACE FIGURE DURING FULL APERTURE
POLISHING," of Tayyab I. Suratwala et al., which is incorporated by
reference herein in its entirety for all purposes.
BACKGROUND OF THE INVENTION
[0003] The present invention generally relates to an apparatus and
a method for shaping an optical surface. More particularly, the
present relates to an apparatus and a method for generating a
deterministic polishing process for an optical surface.
[0004] Optical elements, such as lenses and mirrors, in an optical
system provide for the shaping of radiation fronts, such as light
fronts. Shaping of radiation fronts may include focusing,
culminating, dispersing, and the like. The shapes of the surfaces
of optical elements are one feature of the optical elements that
contribute to shaping radiation fronts as desired. The forming of
optical surfaces of optical elements typically includes a series of
basic process steps including: i) shaping, ii) grinding, iii)
full-aperture polishing, and sometimes iv) sub-aperture polishing.
With significant innovation and development over the years in i)
shaping and iv) sub-aperture polishing, both shaping and
sub-aperture polishing have become relatively deterministic. For
example, with the advent of both computer numerical controlled
(CNC) grinding machines and sub-aperture polishing tools, such as
magnetorheological finishing (MRF), shaping and sub-aperture
polishing have become more deterministic. That is, these processes
may be applied to an optical element, and the resultant surface of
the optical element will have a shape that is desired without
significant human monitoring of the process. For example, a
workpiece (e.g., a fused silica blank) might be placed in a CNC
machine for shaping, and the CNC machine might shape the blank
without the need for a human to stop the CNC machine to change any
of the control parameters of the CNC machine.
[0005] However, the intermediate stages: ii) full aperture grinding
and iii) full aperture polishing are relatively less deterministic
processes. That is, various grinding techniques and polishing
techniques may be applied to an optical element, but to achieve a
desired surface shape, the attention, insight, and intuition of an
optician are typically required to achieve the surface shape
desired. Specifically, grinding techniques and polishing techniques
are often applied to a surface iteratively because measurements of
the surface are made as an optician monitors the applied techniques
and makes adjustments to the techniques. Without the optician's
monitoring and talents, the surfaces of optical elements during
grinding and polishing are highly likely to have a shape that is
not desired. That is, the resultant optical elements might not be
useful for their intended purposes, such as shaping radiation
fronts as desired, or the optical elements might be damaged (e.g.,
in high energy applications) during use due to less than optimal
surface shape.
[0006] The ability to deterministically finish a surface during
full aperture grinding and full aperture polishing will provide for
obtaining a desired surface shape of an optical element in a manner
that is relatively more repeatable, less intermittent, and
relatively more economically feasible than traditional grinding and
polishing techniques. The development of a scientific understanding
of the material removal rate from a surface is one relatively
important step in transitioning to deterministic grinding and
polishing.
[0007] At the molecular level, material removal during glass
polishing is dominated by chemical processes. The most common
polishing media for silica glass is cerium oxide. Cerium oxide
polishing can be described using the following basic reaction:
.dbd.Ce--OH+HO--Si.ident..fwdarw..dbd.Ce--O--Si.ident.+H.sub.2O
(1).
The surface of the cerium oxide particle is cerium hydroxide, which
condenses with the glass surface (silanol surface) to form a
Ce--O--Si bond. The bond strength of this new oxide is greater than
the bond strength of the Si--O--Si bond (i.e., the glass). Hence,
polishing is thought to occur as ceria particles repeatedly tear
away individual silica molecules. It is well known that parameters
such as pH, isoelectric point, water interactions, slurry
concentration, slurry particle size distribution, and other
chemical parameters can influence the removal rate of material from
a surface.
[0008] At the macroscopic level, material removal from a surface
has been historically described by the widely used Preston's
equation:
h t = k p .sigma. o V r ##EQU00001##
where dh/dt is the average thickness removal rate, .sigma..sub.o is
the applied pressure of a lap on a workpiece, and V.sub.r is the
average relative velocity of the polishing particle relative to the
workpiece. The molecular level effects are described
macroscopically by the Preston's constant (k.sub.p). The molecular
level effects include the effects of the particular slurry used for
polishing. As can be seen from Preston's equation, the rate of
removal of material from a surface of a workpiece increases
linearly with pressure .sigma..sub.o and velocity V.sub.r. Many
studies, particularly those in the chemical mechanical polishing
(CMP) literature for silicon wafer polishing, have expanded
Preston's model to account for slurry fluid flow and hydrodynamic
effects, Hertzian contact mechanics, influence of asperity
microcontact, lap bending, and the mechanics of contact on the
pressure distribution. Only a few of these studies focus on
understanding and predicting surface shape (or global
non-uniformity).
[0009] None of the foregoing mentioned studies has described the
general case involving the interplay of these multiple effects such
that the material removal and the final surface shape of the
workpiece can be quantitatively determined. Therefore, new
apparatus and methods are needed to measure and predict material
removal and surface shape for a workpiece (such as a silica glass
workpiece) that has been polished using polishing slurry (such as
cerium oxide slurry) on a lap (such as a polyurethane lap) under a
systematic set of polishing conditions. Further, a spatial and
temporal polishing apparatus and method are needed to simulate the
experimental data incorporating: 1) the friction coefficient as
function of velocity (Stribeck curve), 2) the relative velocity
which is determined by the kinematics of the lap and workpiece
motions, and 3) the pressure distribution, which is shown to be
dominated by: a) moment forces, b) lap viscoelasticity; and c)
workpiece-lap interface mismatch.
BRIEF SUMMARY OF THE INVENTION
[0010] The present invention generally relates to an apparatus and
a method for an offer reporting system. More particularly, the
present invention relates to an apparatus and a method for
generating a deterministic polishing process for an optical
surface.
[0011] One embodiment of the present invention includes a
computerized method for determining an amount of material removed
from a workpiece during a polishing process. The method includes
receiving at a polishing system a set of polishing parameters, and
determining on the polishing system a set of kinematic properties
for a lap and a workpiece of the polishing system from at least a
portion of the set of polishing parameters. The method further
includes determining on the polishing system a time of exposure for
a set of lap points on the workpiece based on at least a portion of
the set of polishing parameters and the set of kinematic
properties, and determining on the polishing system a friction
force between the lap and the workpiece from at least a portion of
the set of polishing parameters. The method further includes
determining on the polishing system a slope between the lap and the
work piece based on a moment force between the lap and the
workpiece, wherein the moment force is based on the determined
friction force, and determining on the polishing system a pressure
distribution between the lap and the workpiece based on a
information for a lap type included in the set of polishing
parameters. The method further includes determining on the
polishing system a cumulative pressure distribution between the lap
and the workpiece based on the slope, the angle, the pressure
distribution for the lap type, and the time of exposure; and
determining on the polishing system an amount of material removed
from the workpiece based on a product of the cumulative pressure
distribution, the friction force, and the set of kinematic
properties.
[0012] According to a specific embodiment of the present invention,
each determining step is executed for a plurality of points on a
surface of the workpiece. The method further includes executing
each determining step for a plurality of successive time
periods.
[0013] According to another specific embodiment, the set of
polishing parameters includes a set of material properties, a set
of polisher configuration parameters, and set of polisher kinematic
properties. The set of material properties includes properties of
the polishing system and includes information for a lap type, a
Stribeck friction curve for the lap, and an optic-lap mismatch. The
set of material properties may further include the Preston's
constant for Preston's equation. The information for the lap type
may be information to identify the lap type as viscoelastic,
viscoplastic, or elastic. The set of polisher kinematic properties
includes a rotation rate of the workpiece, a rotation rate of the
lap, a stroke distance of the workpiece relative to the lap, and a
stroke frequency. The set of polisher configuration parameters
includes a workpiece shape, a lap shape, a workpiece size, a lap
size, a lap curvature, a load distribution of the lap on the
workpiece, and a moment of the workpiece relative to the lap.
[0014] According to another specific embodiment, the method further
includes subtracting the amount of material removed from the
workpiece shape for a first time period to determine a new
workpiece shape for the first time period; and executing each
determining step for a successive time period following the first
time period using the new workpiece shape to determine a successive
amount of material removed from the workpiece for the successive
time period. The method may further include determining a set of
control settings for the polishing system from the new workpiece
shape and a final workpiece shape; and setting on the polishing
system a set of controls to the set of control settings to adjust
the polishing system to polish the workpiece shape to the final
workpiece shape.
[0015] According to another embodiment of the present invention, a
computer readable storage medium contains program instructions
that, when executed by a controller within a computer, cause the
controller to execute a method for determining an amount of
material removed from a workpiece during a polishing process. The
steps of the method are described above.
[0016] According to another embodiment of the present invention, a
computer program product for determining an amount of material
removed from a workpiece during a polishing process on a computer
readable medium includes code for executing the method steps
described above.
[0017] According to another embodiment of the present invention, a
polishing system includes a lap configured to contact a workpiece
for polishing the workpiece, and a septum configured to contact the
lap. The septum has an aperture formed therein to receive the
workpiece, and the lap is configured to contact the workpiece
through the aperture. The polishing system further includes a first
device configured to couple to the workpiece and place a first
amount of pressure between the workpiece and the lap, and a second
device coupled to the septum and configured to place a second
amount of pressure between the septum and the lap to compress the
lap as the workpiece is polished by the lap, wherein the second
amount pressure is three or more times the first amount
pressure.
[0018] According to a specific embodiment of the polishing system,
the compression of the lap is configured to inhibit the workpiece
from compressing the lap as the workpiece is polished by the lap.
The compression of the lap is configured to substantially planarize
the lap as the workpiece is polished by the lap. The polishing
system may further include the workpiece.
[0019] According to another embodiment of the present invention, a
polishing method is provided for pressing a lap with a septum to
compress the lap during polishing of a workpiece to inhibit the
workpiece from compressing the lap during the polishing. The method
includes pressing on a workpiece with a first forcing device to
place a first amount of pressure between a lap and a workpiece; and
pressing on a septum with a second forcing device to place a second
amount of pressure between the septum and the lap, wherein the
septum has an aperture formed therein and the workpiece is
configured to contact the lap through the aperture, and wherein the
second amount of pressure is three or more times greater than the
first amount of pressure. According to a specific embodiment, the
method further includes rotating the lap with respect to the septum
and the workpiece.
[0020] According to another embodiment of the present invention, a
polishing system configured to polish a lap includes a lap
configured to contact a workpiece for polishing the workpiece, and
a septum configured to contact the lap. The septum has an aperture
formed therein. The aperture has substantially the same radius as
the workpiece. The aperture has a center disposed at a radial
distance from a center of the lap, and disposed along a first
radial direction of the lap. The workpiece has a center disposed at
the radial distance from the center of the lap, and disposed along
a second radial direction of the lap.
[0021] According to a specific embodiment of the polishing system,
the septum is configured to polish the lap to a substantially
planar surface as the lap polishes the workpiece. The first radius
and the second radius are oppositely directed. The polishing system
may further include the workpiece. The septum has a substantially
triangular shape.
[0022] These and other embodiments of the present invention are
described in more detail in conjunction with the text below and the
attached figures.
BRIEF DESCRIPTION OF THE DRAWINGS
[0023] FIG. 1 is a simplified block diagram of a polishing system
according to one embodiment of the present invention;
[0024] FIGS. 2A and 2B are a simplified cross-sectional view and a
simplified top view of the set of polishing devices according to
one embodiment of the present invention;
[0025] FIG. 3 is a high-level flow diagram of a computerized method
for generating a set of polishing determinations and a set of
control settings for a set of controls of a polishing system;
[0026] FIG. 4A is a simplified schematic of a viscoelastic lap
deformed by the leading edge of a workpiece passing over the
viscoelastic lap;
[0027] FIG. 4B is a simplified graph of the pressure gradient
across the surface of the workpiece as a function of position on
the surface with respect to a leading edge of the workpiece;
[0028] FIG. 5 is an example graph of a Stribeck friction curve for
a particular lap type, such as a polyurethane lap;
[0029] FIG. 6 is an example schematic of a typical mismatch in
shape between a workpiece and a lap where the workpiece and/or the
lap may have a curved surface;
[0030] The graph at the bottom of FIG. 7 shows the pressure
distribution of the lap on the workpiece due to the frictional
forces for the workpiece moving in the direction of arrow 700;
[0031] FIGS. 8A and 8B are graphs that suggest that increasing the
separation distance, tends to increase the time average velocity
and hence the removal rate of material from the workpiece
surface;
[0032] FIGS. 8C and 8D are graphs that illustrate that increasing
the stroke distance generally leads to lower velocities at the edge
of the workpiece due to the edge of the workpiece spending more
time off of the lap, and hence the workpiece would become more
concave;
[0033] FIG. 9A is a graph that illustrates that the time of lap
exposure can be determined using a line path of some point on the
lap (x.sub.L,y.sub.L) at the leading edge of the workpiece as it
travels to some given point on the workpiece (x,y);
[0034] FIG. 9B is a graph that shows the calculated time of lap
exposure t.sub.L(x,y) for the conditions used for a sample
workpiece;
[0035] FIG. 10 schematically illustrates the delayed elasticity
viscosity model, which is comprised of two moduli (two springs) and
one viscosity (dashpot);
[0036] FIG. 11A shows the calculated pressure distribution using
the conditions described for a sample workpiece where the workpiece
does not rotate;
[0037] FIG. 11B shows the measured surface profile for a sample
workpiece after 1 hour of polishing according to one exemplary
embodiment of the present invention; and
[0038] FIG. 12 is a simplified top view of a polishing system
according to another embodiment of the present invention.
APPARATUS AND METHOD FOR DETERMINISTIC CONTROL OF SURFACE FIGURE
DURING FULL APERTURE PAD POLISHING
Detailed Description of Select Embodiments of the Invention
[0039] The present invention generally provides an apparatus and a
method for shaping an optical surface. More particularly, the
present invention provides an apparatus and a method for generating
a deterministic polishing process for an optical surface.
[0040] FIG. 1 is a simplified block diagram of a polishing system
100 according to one embodiment of the present invention. Polishing
system 100 includes a computer system 105, a set of controls 110,
and a set of polishing devices 115. Polishing system 100 is
configured to polish a workpiece, such as an optical element, as
described below.
[0041] Computer system 105 may be a personal computer, a work
station, a laptop computer, a set of computers, a dedicated
computer, or the like. As referred to herein a set includes one or
more elements. Computer system 105 may include a set of processors
configured to execute one or more computer programs. Computer
system 105 may also include one or more memory devices 120 on which
computer code and any results generated by executing the computer
code may be stored. The one or more memory devices may include one
or more of a RAM, a ROM, a CD and CD drive, an optical drive, etc.
Computer system 105 may also include a monitor 125, and one or more
human interface devices, such as a keyboard 130, mouse 135, a puck,
a joystick, etc. Computer system 105 may be a stand alone computer
system, or may be coupled to the set of controls 110 for
controlling the set of controls to thereby control the polishing of
a workpiece. According to one embodiment, the computer system may
include the set of controls 110. The set of controls may be coupled
to the set of polishing devices, and may be configured to control
the set of polishing devices as described below. According to one
embodiment, computer system 105 is configured to store computer
code and execute computer code to thereby embody various
embodiments of the present invention.
[0042] FIGS. 2A and 2B are a simplified cross-sectional view and a
simplified top view of the set of polishing devices 115 according
to one embodiment of the present invention. The set of polishing
devices 115 includes a base 210, a lap 215, a mounting disk 220, a
driving pin 225, and a viton tube 230. The set of polishing devices
may also include a septum 235. Lap 215 may be a polyurethane lap
and may be coupled to base 210, which may be an aluminum base.
Viton tube 230 is configured to deliver a polishing solution onto
the lap for polishing a workpiece 240. The workpiece may be a
silica glass workpiece and may be attached to mounting disk 220 via
an adhesive 245, such as blocking wax. The polishing solution
supplied by the viton tube may be cerium oxide, which is a
relatively commonly used polishing solution for silica glass. Note
that devices other than a viton tube may be used for delivery a
polishing solution.
[0043] Via a polishing process applied to the workpiece by the
polishing system, a surface 250 of the workpiece disposed adjacent
to the lap may be polished to a desired shape. According to one
polishing embodiment of the present invention, the base and lap may
be rotated by one or more motors 255 in the direction indicated by
arrow 260 at a rotation rate of R.sub.L. The workpiece may be
rotated by the driving pin, which may be coupled to one or more
motors 265 that are configured to rotate the driving pin and
thereby rotate the workpiece. The workpiece may be rotated in a
direction indicated by arrow 270 at a rotation rate of R.sub.O. The
workpiece may also be moved linearly (or stroked) by the driving
pin in the plus and minus x direction through a stroke distance of
plus and minus ds at a stroke rate direct R.sub.S. The driving pin
may be moved linearly by motors 265 or other devices to linearly
move the workpiece. The stroke distance may be measured outward
from a radius S (see FIG. 2B), which is perpendicular to the stroke
direction. The driving pin may also be configured to be moved
vertically up and down along the z axis (up in FIG. 2A, and out
from the page in FIG. 2B) so that a gap may be set between the
workpiece and the lap. As described below, the pressure resulting
between the workpiece and lap is a function of the gap. Various
mechanisms, well known to those of skill in the art, may be
configured to move the workpiece relative to the base for setting
the gap between the workpiece and the lap.
[0044] According to one embodiment, each control in the set of
controls 110 may include a device having a variety of settings for
setting the polishing parameters (R.sub.L, R.sub.O, d.sub.S,
R.sub.S). The gap between the workpiece and the lap is described
above. The set of controls may include knobs, sliders, switches,
computer activated controls, and the like. According to one
embodiment, in which computer system 105 includes the set of
controls, the controls may be on-screen controls displayed on the
computer monitor. The on-screen controls may control program code
and computer interfaces for controlling the set of polishing
parameters.
[0045] FIG. 3 is a high-level flow diagram of a computerized method
300 for generating a set of polishing determinations 305 and a set
of control settings 310 for the set of controls 110 according to
one embodiment of the present invention. Each polishing
determination in the set of polishing determinations 305 is labeled
in FIG. 3 with the base reference number 305 and an alphabetic
suffix. It should be understood that the high-level flow diagram is
exemplary. Those of skill in the art will understand that various
steps in the method may be combined and addition steps may added
without deviating from the spirit and purview of the described
embodiment. The high-level flow diagram is not limiting on the
claims. Computerized method 300 is first described in a high-level
overview, and then is described in further detail thereafter.
Computerized method 300 may be executed on polishing system 100.
More specifically, many of the steps of computerized method 300 may
be executed on the polishing system's computer system 105.
[0046] In high level overview, computerized method 300 simulates a
polishing process on polishing system 100. The output of the
computerized method includes a prediction for a shape of a surface
of a workpiece under a set of polishing conditions, and a
prediction for the set of control settings 310 for the set of
controls 110. The shape of a surface of a workpiece is sometimes
referred to herein as a surface figure. According to one embodiment
of the present invention, computer system 105 is configured to
receive a set of polishing parameters 315 (labeled 315a, 315b, and
315c) for a polishing process of a workpiece and iteratively
determine the amount of material removed from the workpiece.
Computer system 105 may also be configured to use the polishing
parameters to determine the shape of the surface of the workpiece
305a, the pressure distribution between the workpiece and lap 305b,
the time averaged velocity for the workpiece relative to the lap
305c, the amount of time the workpiece is exposed to the lap 305d,
the shape of the surface of the lap 305e, the removal rate of
material from the workpiece 305f, the slope of the workpiece
relative to the lap 305g, and/or the like.
[0047] The set of polishing determinations 305 may be generated for
a set of points on the workpiece and the lap. The set of polishing
determinations may be for a set of successive time periods
.DELTA.t.sub.1, .DELTA.t.sub.2, .DELTA.t.sub.3 . . .
.DELTA.t.sub.n. The set of points may include hundreds, thousands,
tens of thousands, or more points on the workpiece and/or lap. The
temporal length of the time periods .DELTA.t may be set as desired.
For each latest time period .DELTA.t, the amount of material
determined to be removed in the immediately prior time period
.DELTA.t is used by the computer system to determine the subsequent
amount of material removal. That is, the computerized method uses
the method's output (e.g., polishing determinations 305) as the
input to the computerized method for successive temporal steps
.DELTA.t. Based on the amount of material determined to be removed
at each time period .DELTA.t, the set of control settings 310 may
be determined by computer system 315. A human user or computer
system 105 may use the set of control settings 310 to set the set
of controls 110 on polishing system 100.
[0048] According to one embodiment, computer system 105 is
configured to store and execute computer code in the form of a
polishing model, which is configured to receive the set of
polishing parameters 315 to generate the set of polishing
determinations 305 and generate the set of control settings 310.
According to one embodiment, the polishing model is a modified
Preston's model shown in equation 1 below.
h i ( x , y , t ) t = k p .mu. ( v r ( x , y , t ) ) .sigma. o ( x
, y , t ) v r ( x , y , t ) ##EQU00002##
The modified Preston's model is both a spatial and temporal model.
The modified Preston's model takes into account the kinematics
between the workpiece and the lap, and the nonuniformities in the
pressure distribution between the workpiece and the lap. Both the
kinematics and the nonuniformities in pressure may be empirically
and/or theoretically determined and may be used in the modified
Preston's model.
[0049] In the modified Preston's model,
h i ( x , y , t ) t ##EQU00003##
is the instantaneous removal rate of material from a workpiece, at
a given time t and a given position (x,y) on the workpiece.
.mu.(.nu..sub.r(x,y,t)) is the friction coefficient between the
workpiece and the lap. The friction coefficient is a function of
the relative velocity .nu..sub.r(x,y,t) between the workpiece and
the lap at the workpiece-lap interface. .sigma..sub.o(x,y,t) is the
pressure distribution resulting from the applied pressure
(.sigma..sub.o) and the characteristics of the workpiece-lap
contact. k.sub.p is the Preston's constant, which is a fundamental
removal rate of material from the workpiece for a given polishing
compound (e.g., ceria slurry). More specifically, the Preston's
constant is the removal rate of material from the workpiece per
unit pressure between the workpiece and the lap and the unit
velocity between the points on the workpiece and the lap.
[0050] According to one embodiment, the method shown in FIG. 3, for
determining material removal from the surface of a workpiece and
determining settings for the controls of the polishing system, is
based on the modified Preston's equation. The modified Preston's
equation takes into account the empirically measured and/or
theoretically determined effects of: 1) the frictional forces
between the workpiece and the lap as function of relative velocity
between the polishing particle and workpiece; 2) the relative
velocity between the workpiece and lap based on various kinematics;
and 3) the factors that affect the pressure distribution between
the workpiece and the lap (such as, moment forces and workpiece
tilt, lap viscoelasticity, and workpiece-lap interface mismatch).
These effects are combined to generate the method shown in FIG. 3
and to generate a more global material removal model.
[0051] As described briefly above, the material removal and shape
of a surface of a workpiece after polishing (e.g., ceria pad
polishing) have been measured and analyzed as a function of
kinematics, loading conditions, and polishing time. Also, the
friction at the workpiece-lap interface, the slope of the workpiece
relative to the lap plane, and lap viscoelastic properties have
been measured and correlated to material removal. The results show
that the relative velocity between the workpiece and the lap (i.e.
the kinematics) and the pressure distribution determine the spatial
and temporal material removal, and hence the final surface shape of
the workpiece. In embodiments where the applied loading and
relative velocity distribution over the workpiece are spatially
uniform, a significant non-uniformity in material removal, and thus
surface shape, is observed. This is due to a non-uniform pressure
distribution resulting from: 1) a moment caused by a pivot point
and interface friction forces; 2) viscoelastic relaxation of the
polyurethane lap; and 3) a physical workpiece-lap interface
mismatch. For completeness, both the kinematics and the
non-uniformities in the pressure distribution are described below
as the steps of computerized method 300 are described in further
detail.
[0052] The high-level flow chart for the computerized method 300
shown in FIG. 3 is described in further detail immediately below.
At a step 320, the computer system is configured to receive a set
of material properties 315a for polishing system 100. The set of
material properties 315a may be received by computer system 105
from a local memory, a remote memory on a network or the like. The
material properties may include information for i) a lap type being
used in polishing system 100, ii) a Stribeck friction curve, iii) a
workpiece-lap mismatch response, and iv) Preston's constant
(k.sub.p). Each of material properties 315a is described in detail
below.
[0053] At a step 325, the computer system is configured to receive
a set of configuration properties 315b for a configuration of
polishing system 100. The set of configuration properties 315b may
be received by computer system 105 from a local memory, a remote
memory on a network or the like. The set of configuration
properties may include: i) the workpiece shape and the lap shape,
ii) the workpiece size and the lap size, iii) the lap curvature,
iv) the load and load distribution of the lap against the
workpiece, and v) the moment of the workpiece relative to the lap.
Each of configuration properties 315b is described in detail
below.
[0054] At a step 330, the computer system is configured to receive
a set of kinematic properties 315c for polishing system 100. The
set of kinematic properties 315c may be received by computer system
105 from a local memory, a remote memory on a network or the like.
The set of kinematic properties 315c may include: i) the rotation
rate R.sub.L of the lap, ii) the rotation rate R.sub.O of the
workpiece, iii) the stroke length d.sub.S of the workpiece, the
stroke frequency R.sub.S. The kinematic properties are generally
well known by those of skill in the art.
Material Properties
[0055] As described briefly above, the set of material properties
315a may include: i) a lap type being used in polishing system 100,
ii) a Stribeck friction curve, iii) a workpiece-lap mismatch
response, iv) lap type wear rate, and iv) Preston's constant
(k.sub.p). According to one embodiment of the present invention,
the information for the lap type may include information that
identifies the lap as an elastic lap, a viscoelastic lap, a
viscoplastic lap, or other lap type. Viscoelasticity in general is
the property of materials that exhibit both viscous and elastic
characteristics if deformed. A viscoelastic lap may be deformed
(e.g., compressed) by an applied force, and after removal of the
applied force or a reduction of the applied force, the molecules in
the viscoelastic lap may relax and expand from the deformation.
More specifically, viscous materials tend to resist shear flow and
strain linearly with time if a stress is applied to the material.
Elastic materials strain instantaneously when stretched and just as
quickly return to their original state once the stress is removed.
Viscoelastic materials have elements of both of these properties
and, as such, exhibit time dependent strain.
[0056] FIG. 4A is a simplified schematic of a viscoelastic lap
(such as a polyurethane lap) deformed by the leading edge of the
workpiece passing over the viscoelastic lap. Across the workpiece
surface 250, the leading edge 410 of the workpiece is exposed to
the highest pressure by the lap as the workpiece moves across the
workpiece in the direction 415. As the lap relaxes from being
deformed there may be a pressure gradient applied to the workpiece
as the workpiece moves relative to the lap. FIG. 4B is a simplified
graph of the pressure gradient across the surface of the workpiece
as a function of position on the surface with respect to leading
edge 410. The highest pressure applied to the workpiece is at the
leading edge 410 and drops away from the leading edge. At
subsequent steps in computerized method 300, this pressure gradient
on the surface of the workpiece is combined with other pressure
effects and pressure information to determine a cumulative pressure
across the surface of the workpiece.
[0057] FIG. 5 is an example graph of a Stribeck friction curve for
a particular lap type, such as a polyurethane lap. A Stribeck
friction curve provides the friction coefficient between the
workpiece and the lap based on: i) the applied pressure between the
workpiece and the lap, and ii) the relative velocity between the
workpiece and the lap at each point on the workpiece and lap. The
friction between the workpiece and the lap generally decreases with
increased velocity between the workpiece and the lap as shown in
FIG. 5. The friction between the workpiece and the lap generally
increases with increased pressure between the workpiece and the
lap. The Stribeck friction curve may be a function of the slurry.
The Stribeck friction curve may be determined empirically for a
lap.
[0058] In general, the contribution of interfacial friction to
material removal (see equation 1 above) can be thought of as being
proportional to the number of polishing particles making contact
with the workpiece. The greater the number of particles making
contact with the surface of the workpiece, the greater the
friction, and the greater the removal rate of material from the
surface. According to one embodiment of the present invention, the
friction force (F) was measured as a function of applied load (P)
and lap rotation rate (R.sub.L). The friction coefficient (.mu.)
for each measurement is then: .mu.=F/P. The magnitude of the
friction between the workpiece and the lap may be determined by the
mode of contact between the workpiece and the lap, the applied
load, the characteristics of the slurry (e.g. viscosity), and the
workpiece to lap relative velocity. It is common to describe
dynamic friction coefficient .mu. as function of
.eta. s v r .sigma. o ##EQU00004##
where .eta..sub.s is the slurry fluid viscosity. Note that the
friction coefficient can change relatively significantly depending
on the velocity and applied pressure. At relatively low values
of
.eta. s v r .sigma. o ( e . g . , < 10 - 6 m ) ##EQU00005##
for the lap, the workpiece and the lap make mechanical contact
(referred to as contact mode), and the friction coefficient is
relatively high (0.7-0.8). At relatively high values of
.eta. s v r .sigma. o ( e . g . , > 10 - 5 m ) ,
##EQU00006##
the fluid pressure of the slurry carries the workpiece off of the
lap (referred to as hydrodynamic mode), and the friction
coefficient is relatively low (<0.02). Most conventional optical
polishing is performed in contact mode, where the friction
coefficient is large and does not significantly change. Notice in
FIG. 5 that the polyurethane lap, pitch, and the IC1000 pad follow
the same basic behavior with the friction coefficient on the
Stribeck curve. However, the transition into hydrodynamic mode
occurs at different values of
.eta. s v r .sigma. o ##EQU00007##
depending, for example, on the properties of the lap material. For
the polyurethane pad, the friction coefficient can be described by
a sigmoidal curve, which is often used to describe the shape of the
Stribeck curve, as:
.mu. = 0.7 - 0.6 1 + ( 7.7 .times. 10 4 m - 1 n s v r .sigma. o ) .
2 ##EQU00008##
According to one embodiment, the above equation 2 for the friction
coefficient is used in the modified Preston's equation along with
other terms described below to predict the surface shape of a
workpiece and to determine the set of control settings for the set
of controls for the polishing system.
[0059] FIG. 6 is an example schematic of a typical mismatch 600 in
shape between a workpiece and a lap where the workpiece and/or the
lap may have a curved surface. FIG. 6 also shows the workpiece-lap
mismatch response 605 between the workpiece and the lap for the
given mismatch 600. The workpiece-lap mismatch response, in
general, is the pressure variation across the surface of the
workpiece on the lap due to the mismatch in the surface shapes of
the workpiece and the lap. Generally the pressure between the
workpiece and the lap is greatest where the workpiece and/or the
lap have a surface portion that project towards the other. As can
be seen in the exemplary workpiece-lap mismatch response 605, the
pressure is greatest between the workpiece and the lap towards the
outside 610 of the workpiece where the surface of the workpiece has
a maximum surface extension towards the lap. The workpiece-lap
mismatch response may be determined based on a number of factors,
such as variously shaped mismatches, the elasticity of the lap, and
the like. As will be described below, the workpiece-lap mismatch
response may be combined with other pressure information, to
generate a pressure map for the surfaces of the workpiece and the
lap.
Configuration Properties
[0060] As described briefly above, the set of configuration
properties 315b may include: i) the workpiece shape and the lap
shape, ii) the workpiece size and the lap size, iii) the lap
curvature, iv) the load and load distribution of the lap against
the workpiece, and v) the moment of the workpiece relative to the
lap. The configuration properties generally describe certain
aspects of how the set of polishing devices 115 are arranged.
[0061] According to one embodiment of the present invention, the
workpiece shape supplied to computer system 105 includes
information for the flatness and/or the curvature of the surface of
the workpiece prior to polishing. Similarly, the lap shape supplied
to computer system 105 includes information for the flatness of the
surface of the lap prior to polishing. The workpiece size supplied
to the computer system includes the size, e.g., the radius, of the
workpiece that is to be polished, and the lap size is the size,
e.g., the radius, of the lap. The lap curvature supplied to
computer system 105 includes information for the surface curvature
of the lap. The load and the load distribution include information
for the load and load distribution applied to the workpiece, for
example by the driving pin, and/or the lap.
[0062] The moment force information supplied to computer system 105
describes a force that tends to tilt the workpiece relative to the
lap. The moment force arises from the frictional forces on the
workpiece while the workpiece is in motion relative to the lap.
Information for the moment force provided to computer system 105
may include information for the moment force and/or the pressure
distribution across the surface of the workpiece from the moment
force. FIG. 7 is a simplified schematic of the workpiece under a
moment force from the frictional forces. The graph at the bottom of
FIG. 7 shows the pressure distribution of the lap on the workpiece
due to the frictional forces for the workpiece moving in the
direction of arrow 700.
[0063] A moment force driven by the friction between the workpiece
and the lap interface while in contact mode is described. Consider
the workpiece-lap setup as shown in FIGS. 2A and 2B where the
workpiece is held by a spindle and allowed to rotate. Using a force
and moment balance while at equilibrium, the total load and moment
are given by:
P = .intg. opic .sigma. ( x , y ) x y 3 M x = .intg. opic .sigma. (
x , y ) y x y - F y d = 0 4 M y = F x d - .intg. opic .sigma. ( x ,
y ) x x y = 0 5 ##EQU00009##
where F.sub.x and F.sub.y are the components of the friction force
and M.sub.x and M.sub.y are the moment in the x and y direction.
Referring again to FIG. 7, this figure shows the result for
workpiece slope during polishing. The slope increases (where the
leading edge of the workpiece is lower than the trailing edge) with
moment arm distance and applied pressure. This is qualitatively
consistent with the above formalism, since it would result in
higher pressure at the leading edge of the workpiece. The
determined moment and slope (determined using the load and moment
equations shown above) becomes more complicated with the addition
of stroke in the kinematics where the moment and hence slope become
time dependent (i.e. slope changes with position of the workpiece
along the stroke trajectory). Also, any offset of the workpiece
from the lap surface changes the pressure distribution over a
smaller area of the workpiece, and any offset of the workpiece from
the lap surface can also lead to an additional slope due to a
center of gravity balance. The slope due to the moment combined
with the viscoelastic lap contributions lead to a non-uniform
pressure distribution.
[0064] Referring again to FIG. 3, at a step 335, computer system
105 is configured to calculate the position and velocity for each
point on the workpiece as a function of time relative to the points
on the lap (generally referred to as kinematics). The calculation
at step 335 is carried out based on the set of kinematic properties
315c received by the computer system at step 330.
[0065] Material removal from a workpiece is a function of kinematic
properties 315c. See equation 1 above. One of the kinematic
properties that effect material removal from a workpiece is the
relative velocity between the lap surface and the workpiece
surface. The kinematics of the relative velocity of a polishing
particle to the workpiece is described in further detail
immediately below. Polishing particles having relatively high
velocity typically provide for a relatively larger number of the
polishing particles interacting with the workpiece surface, thus
leading to greater material removal per unit time. Assuming that
the workpiece-particle relative velocity is roughly equivalent to
the workpiece-lap relative velocity (i.e., the polishing particle
is essentially stationary relative to the lap), the kinematic
parameters of the system may be used to calculate the relative
velocity of the polishing particles for all points on the
workpiece. It is convenient to describe the relative velocity in
vector form as:
v r ( x , y , t ) = ( R 0 .times. .rho. 0 ( x , y , t ) ) - ( R L
.times. .rho. 0 ( x , y , t ) - S ( t ) ) + S ( t ) t 6
##EQU00010##
where .rho..sub.o is a position on the workpiece given by
coordinates x and y with the origin at the workpiece center, {right
arrow over (R)}.sub.0 and {right arrow over (R)}.sub.L are the
rotation rates of the workpiece and lap in vector form directed
along the z-axis, and {right arrow over (S)} is the vector
describing the separation between the geometric centers of the
workpiece and lap (see FIGS. 2A and 2B). The first term on the
right hand side of equation 6 describes the rotational velocity of
the workpiece for some given position on the workpiece at the
workpiece-center frame of reference. The second term on the right
hand side of equation 6 describes the rotational velocity of the
lap at the workpiece-center frame of reference. The final term on
the right hand side of equation 6 describes the relative velocity
due to the linear motion of the stroke. For a spindle polishing
embodiment (e.g., polishing system 100), each of the terms above
may be described in vector form as:
R o = ( 0 0 R o ) 7 R o = ( 0 0 R L ) 8 S = ( d s sin ( R s , t ) s
0 ) 9 .rho. o = ( x 2 + y 2 sin ( arctan x / y ) + 2 .pi. R o t x 2
+ y 2 cos ( arctan x / y ) + 2 .pi. R o t 0 ) 10 ##EQU00011##
[0066] In order to describe a typical continuous polisher (CP),
d.sub.s is set equal to 0. Since the relative velocity between the
workpiece and a polishing particle can only lead to removal when
the lap and workpiece are in contact, an additional condition for a
non-zero relative velocity applies for the case of a circular
lap:
|{right arrow over (.rho.)}.sub.o(x,y,t)-{right arrow over
(S)}(t).ltoreq.r.sub.L. 11
[0067] The time average relative velocity is then given by:
V r ( x , y ) = 1 t .intg. 0 t v r ( x , y , t ' ) t ' 12
##EQU00012##
[0068] Using equations 6-12, the time average velocity may be
calculated for a variety of kinematics as shown in FIGS. 8A-8D
where r.sub.o=0.05 m, r.sub.L=0.10 m, RL=28 rpm. When V.sub.r is
higher on the edge relative to the center, the workpiece generally
would become convex, and when Vr is lower on the edges, the
workpiece would become concave. FIG. 8A suggests that as the
workpiece rotation rate is mismatched from the lap rotation rate,
the workpiece would generally become more convex. FIGS. 8A and 8B
suggest that increasing the separation distance tends to increase
the time average velocity, and hence the removal rate of material
from the workpiece surface. FIGS. 8C and 8D illustrate that
increasing the stroke distance generally leads to lower velocities
at the edge of the workpiece due to the edge of the workpiece
spending more time off of the lap, and hence the workpiece would
become more concave. These trends are consistent with those
generally observed by opticians during conventional polishing.
[0069] Referring again to FIG. 3, at a step 340, the time of
exposure of each point on the lap to the workpiece is calculated.
More specifically, a point on the lap initially makes contact with
the workpiece at one side of the workpiece (e.g., the leading edge
of the workpiece based on the direction of travel of the workpiece
relative to the lap), the point on the lap travels under the
workpiece and then comes out from under the workpiece where the
point no longer makes contact with the workpiece. This exposure
time for each point on the lap is calculated based on the
kinematics calculated in step 335 and the lap properties, such as
the viscoelastic prosperities. The viscoelastic properties of the
lap and the time of exposure (based on the viscoelastic properties
of the lap) are described in detail immediately below. According to
one embodiment of the present invention, the exposure time may be
used to determine the pressure distribution of the lap on the
workpiece (described immediately below).
[0070] For a viscoelastic lap loaded by an elastic workpiece, the
pressure distribution on the workpiece (.sigma.(x,y)) can be
described by the heredity equation for a constant applied load
as:
.sigma. ( x , y ) = .intg. 0 t L ( x , y ) E rel ( t L ( x , y ) -
t ' ) . ( t ' ) t ' 13 ##EQU00013##
where t.sub.L(x,y) is the time of lap exposure at some point (x,y)
on the workpiece for the corresponding point on the lap, E.sub.rel
is the stress relaxation function for the viscoelastic lap
material, and .epsilon.(t') is the lap strain rate. Each of these
three parameters is analytically described below.
[0071] The time of lap exposure can be determined using a line path
of some point on the lap (x.sub.L,y.sub.L) at the leading edge of
the workpiece as it travels to some given point on the workpiece
(x,y) as illustrated in the schematic in FIG. 9A. For the case of
kinematics without stroke, the time of lap exposure is given
by:
t L ( x , y ) = 1 R L arccos ( x x L ( x , y ) + ( y + s ) ( y L (
x , y ) + s ) x 2 + ( y + s ) 2 ) 14 y L ( x , y ) = x 2 + ( y + s
) 2 - r 0 2 - s 2 15 x L ( x , y ) = r 0 2 - y L ( x , y ) 2 16
##EQU00014##
[0072] Note for every point selected on the workpiece (x,y), there
is a unique corresponding point at the leading edge of the
workpiece (x.sub.L,y.sub.L). FIG. 9B plots the calculated time of
lap exposure t.sub.L(x,y) for the conditions used for a sample
workpiece using the above three equations 1-3. The minimum time of
lap exposure is at the leading edge of the workpiece and the
maximum time of exposure is at the trailing edge on the side of the
workpiece closest to the lap center. The asymmetry of the time of
lap exposure is due to the fact that the velocity of a given point
on the lap is lower closest to the lap center, which leads to
longer times of lap exposure. For the example embodiment shown in
FIG. 9B, the maximum time of lap exposure is 0.6 sec. A similar
exercise, as described above, can be performed for the case with
stroke added; however, the algebra is more complicated. Also, the
time of lap exposure would change along the stroke cycle, whereas
without stroke the time of lap exposure stays constant. The
viscoelastic lap behavior can be modeled using a delayed elasticity
viscosity model described in the known literature. FIG. 10
schematically illustrates the delayed elasticity viscosity model,
which is comprised of two moduli (two springs) and one viscosity
(dashpot). The creep compliance function J(t) and the stress
relaxation function E.sub.rel(t) for the delayed elasticity
viscosity model are described as:
J ( t ) = 1 E 1 + 1 E 2 ( 1 - - t .tau. c ) 17 E rel ( t ) = E 1 E
1 + E 2 ( E 2 + E 1 - t .tau. s ) 18 ##EQU00015##
where .tau..sub.c is the creep compliance time constant and
.tau..sub.s is the stress relaxation time constant for the lap. For
this model the following self similar relationships apply:
E 1 + E 2 = E 19 .tau. c = .eta. E 2 20 .tau. s = .eta. E 21
##EQU00016##
where E and .eta. are the bulk modulus and viscosity of the lap.
This simple viscoelastic model (delayed elasticity model) is one
possible viscoelastic model according to one embodiment of the
present invention. According to other embodiments of the invention,
other more complex, possibly more realistic models are considered
for implementation.
[0073] According to one embodiment of the present invention, from
dynamic mechanical analysis performed on a sample polyurethane lap,
E=100 MPa and .eta.=9.7.times.107 poise. Hence using equations 19,
20, and 21, E.sub.1=97.75 MPa, E.sub.2=2.25 MPa and .tau..sub.s=0.1
sec. Note that the stress relaxation time constant (.tau..sub.s) is
less than the maximum time of lap exposure (see FIG. 9B),
suggesting that a significant amount of stress relaxation can occur
under these set of kinematics with this pad. With all of these
parameters quantitatively known, the stress relaxation function
(equation 13) is quantitatively defined.
[0074] The final component used to determine the pressure
distribution (using equation 13) due to viscoelastic relaxation is
the strain rate (.epsilon.(t'))). The strain on the lap is
constrained by the shape of the workpiece and its orientation with
respect to the lap (i.e., the slope). For cases where the workpiece
surface is flat, the strain as a function of workpiece position can
then be defined as:
( x , y ) = tan ( .theta. x ) x t pad + tan ( .theta. y ) y t pad +
0 22 ##EQU00017##
where .theta..sub.x and .theta..sub.y are the slopes of the
workpiece in the x and y directions relative to the lap plane,
.epsilon..sub.o is the elastic strain at the center of the
workpiece, and t.sub.pad is the thickness of the viscoelastic pad.
It is convenient to describe the strain as a function of time
(.epsilon.(t)) instead of position, which can be done using:
x = r arc cos ( R L t + ( arccos x L r arc ) ) 23 y = r arc sin ( R
L t + ( arccos x L r arc ) ) - s 24 r arc = x 2 + ( y + s ) 2 25
##EQU00018##
where r.sub.arc is the arc radius for a given point (x.sub.L,
y.sub.L) at the leading edge of the workpiece (see FIG. 9A)
relative to the lap center. Substituting into equation 22, and then
differentiating, gives the strain rate as:
( .tau. ) = - tan ( .theta. x ) t pad r arc sin ( R L t + ( arccos
x L r arc ) ) - tan ( .theta. y ) t pad r arc cos ( R L t + (
arccos x L r arc ) ) 26 ##EQU00019##
[0075] Using equation 13-22, the pressure distribution on a
non-rotated workpiece may be determined. FIG. 11A shows the
calculated pressure distribution using the conditions described for
a sample workpiece where the workpiece does not rotate. For
comparison, the measured surface profile for the sample workpiece
after 1 hour of polishing is shown in FIG. 11B. Note the leading
edge of workpiece in each image is designated by a star. The
observed removal is qualitatively consistent with the calculated
pressure distribution where the leading edge experiences a much
higher removal or pressure. For all of the other samples examined,
the workpiece was rotated. Hence the average pressure distribution
may be a time-average of the non-rotated pressure distribution
rotated about the center of the workpiece, which can be described
as:
.sigma. ( r ) = 1 2 .pi. .intg. 0 2 .pi. .sigma. ( r , .theta. )
.theta. 27 ##EQU00020##
where .sigma.(r,.theta.) is the pressure distribution determined by
equation 13 above in cylindrical coordinates. As the slope of the
workpiece is increased relative to the lap plane, in equation 26,
the time average rotated pressure distribution becomes more
non-uniform, and hence the material removal becomes more
non-uniform.
[0076] Referring again to FIG. 3, at a step 345 the friction, for
the current time period .DELTA.t, at each point on the workpiece is
determined based on the kinematics determined in step 335, and the
Stribeck curve received by computer system 105 at step 320. The
friction is a function of velocity of each point on the workpiece
relative to the lap. The friction at a point determines the amount
of material removal at the point. At step 345, the moment force on
the workpiece is also determined. The determination of the moment
force also provides angle of the workpiece relative to the lap. The
angle between the workpiece and the lap effects the pressure
distribution between the workpiece and the lap.
[0077] At a step 350, for the current time period .DELTA.t, the
slope between the workpiece and the lap are determined from the
angle between the workpiece and the lap determined in step 345. At
step 350, the pressure distribution from the slope of workpiece
relative to the lap is also determined.
[0078] At a step 355, for the current time period .DELTA.t, the
pressure distribution based on the type of lap specified in step
320 is determined. For example, step 355a is executed if the lap is
an elastic lap. For an elastic lap the rigid punch pressure
distribution is determined. Step 355b is executed if the lap is a
viscoelastic lap. For a viscoelastic lap, based on the exposure
time determined in step 340, the viscoelastic pressure distribution
of the lap against the workpiece is determined for each point on
the workpiece. Sometimes pressure distribution is referred to
herein as stress distribution. The relaxation of the lap at each
point on the workpiece is also determined. Step 355c is executed if
the lap is a viscoplastic lap. For a viscoplastic lap, the
viscoplastic pressure distribution of the lap against the workpiece
is determined for each point on the workpiece. The permanent
deformation for all points on the lap is also determined for a
workpiece pressing into the lap. The permanent deformation is a
plastic deformation due to the plastic properties of the lap.
[0079] At a step 360, for the current time period .DELTA.t, the
"cumulative" pressure of the lap on the workpiece is determined for
all points on the workpiece as the workpiece moves relative to the
lap. The cumulative pressure distribution is determined based on
each of the pressure distributions, as described above, including
the pressure distribution effects from the specific lap type (step
355), the pressure distribution from the workpiece-lap mismatch,
and the pressure distribution from the slope between the workpiece
and the lap, the pressure distribution from the lap curvature,
and/or the pressure distribution from the lap deflection. The
cumulative pressure distribution on the lap may be the product of
the discrete pressure distributions from the various physical
phenomena where each phenomenon has its own pressure distribution
as described above. At a step 365, the cumulative pressure of the
lap on the workpiece is normalized.
[0080] At a step 370, for the current time period .DELTA.t, the
total material removal at each point on the workpiece is determined
based on the modified Preston's equation
h i ( x , y , t ) t = k p .mu. ( v r ( x , y , t ) ) .sigma. o ( x
, y , t ) v r ( x , y , t ) . 28 ##EQU00021##
(described in detail above), where the friction coefficient
.mu.(.nu..sub.r(x,y,t)) is determined for each point on the
workpiece at step 345, the cumulative pressure distribution
.sigma..sub.o(x,y,t) of the lap on the workpiece is determined for
each point on the workpiece at steps 360 and 365, and the relative
velocity .nu..sub.r(x,y,t) for each point on the workpiece relative
to the lap determined at step 335.
[0081] At a step 375, based on the amount of material removal
determined at step 370 and the initial known surface shape of the
workpiece supplied to computer system 105 at step 325, a new
surface shape of the workpiece may be determined by computer system
105 for each point on the workpiece, for example by simple
subtraction. According to one embodiment of the present invention,
the steps of the computerized method shown in FIG. 3 may be
repeated one or more times using the newly determined surface shape
of the workpiece to calculate total material removal across the
surface of the workpiece for one or more subsequent time periods
.DELTA.t.
[0082] According to one embodiment of the present invention, after
a given number of time periods .DELTA.t, the surface shape of the
workpiece determined at step 375 is compared to the desired-final
surface shape of the workpiece. Based on the difference between the
surface shape determined at step 375 and the desired-final surface
shape, the set of control settings 310 for the set of controls 110
may be determined. For example, the set of control settings may be
for changing the load on the workpiece, changing the workpiece
rotation rate, the lap rotation rate, the stoke length, the stroke
rate or the like.
[0083] At step 375, computer system 105 may be configured to
determine other operating parameters, save the operating
parameters, and/or report (e.g, display on the computer monitor)
the operating parameters of the polishing devices 115. For example,
the surface shape of the lap may be determined as the surface shape
changes during polishing. The modified Preston's equation may be
applied to the lap to determine material removal for the lap for
one or more successive time periods .DELTA.t. According to a
further example, the cumulative pressure distribution may be
determined, the time average velocity for each point on the
workpiece may be determined, the time that each point on the
workpiece is exposed to the lap may be determined. A material
removal rate for the workpiece and/or the lap may be
determined.
Lap Pre-Compression
[0084] According to another embodiment of the present invention,
lap 215 is pre-compressed during polishing to flatten the lap
surface. Pre-compressing the lap surface reduces the compression of
the lap caused by the workpiece moving with respect to the lap.
Reducing the amount of lap compression caused by the workpiece
moving relative to the lap provides that the pressure distribution
of the lap on the workpiece is relatively more uniform than the
pressure distribution of a lap that is not pre-compressed.
According to one embodiment, the lap is pre-compressed by placing
pressure on septum 235 (see FIG. 2A) to thereby place pressure on
the lap for pre-compression. According to one embodiment of the
present invention, the unit pressure of septum 235 on lap 215 is
three or more times the amount of the unit pressure of the
workpiece on the lap. The septum may be pressed into the lap by one
or more of a variety of devices. Those of skill in the art will
know of forcing devices that may be coupled to the septum where the
forcing device may be configured to press the septum into the lap
at the above discussed unit pressure. According to one embodiment
of the present invention, the septum is glass.
Lap Polishing
[0085] FIG. 12 is a simplified top view of a polishing system 1200
according to another embodiment of the present invention. Polishing
system 1200 differs from polishing system 100 described above in
that polishing system 1200 includes a septum 1205, which may not
surround the workpiece. Septum 1205 may be generally triangular in
shape as viewed from the top of the septum, and relatively planar
as viewed from the side. Specifically, the septum may have first
and second sides 1206 and 1207, respectively, which are relatively
straight as viewed from the top of the septum as shown in FIG. 12.
The first and second sides may join at an apex 1208. Apex 1208 may
be configured to be at a center of the lap. The septum may further
include a curved side 1209 as viewed from the top. The curved side
may have a radius of curvature, which might match a radius of
curvature of the lap. Septum 1205 may have an opening 1210 formed
therein. Opening 1210 may have a radius that is substantially the
same as the radius of the workpiece. The center of opening 1210 and
the center of the workpiece 240 may be at substantially the same
radial distance from the center of the lap 215, but may lie along
different radius of the lap. According to one specific embodiment,
septum 1205 may be positioned on the lap substantially opposite to
the workpiece (i.e., on oppositely pointing radius of the lap) That
is, the center of opening 1210 and the center of the workpiece may
lie on the substantially same diameter of the lap. The inventors
have discovered that a roughly triangular shaped septum polishes
the lap in a relatively uniform manner as the workpiece is
polished. Polishing the lap in a relatively uniform manner as the
workpiece is polished provides that the uneven pressure
distributions from the lap wearing in a non-uniform manner are
lowered. According to one embodiment, polishing system 1200 does
not include septum 235 as shown in FIG. 2A.
[0086] It is to be understood that the examples and embodiments
described above are for illustrative purposes only and that various
modifications or changes in light thereof will be suggested to
persons skilled in the art, and are to be included within the
spirit and purview of this application and scope of the appended
claims. Therefore, the above description should not be understood
as limiting the scope of the invention as defined by the
claims.
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